{"title":"The \\(L^p\\) Estimate for the Gain Term of the Boltzmann Collision Operator and Its Application","authors":"Ling-Bing He, Jin-Cheng Jiang, Hung-Wen Kuo, Meng-Hao Liang","doi":"10.1007/s00205-024-02067-8","DOIUrl":null,"url":null,"abstract":"<div><p>We prove the Hardy–Littlewood–Sobolev type <span>\\(L^p\\)</span> estimates for the gain term of the Boltzmann collision operator including Maxwellian molecule, hard potential and hard sphere models. Combined with the results of Alonso et al. (Comm Math Phys 298: 293–322, 2010) for the soft potential and Maxwellian molecule models, we provide a unified form of <span>\\(L^p\\)</span> estimate for all cutoff models which are sharp in the sense of scaling. The most striking feature of our new estimates for the hard potential and hard sphere models is that they do not increase the moment, the same as Maxwellian molecule and soft potential models. Based on these novelties, we prove the global existence and scattering of the non-negative unique mild solution for the Cauchy problem of the Boltzmann equation when the positive initial data is small in the weighted <span>\\(L^3_{x,v}\\)</span> space.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6000,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archive for Rational Mechanics and Analysis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00205-024-02067-8","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
We prove the Hardy–Littlewood–Sobolev type \(L^p\) estimates for the gain term of the Boltzmann collision operator including Maxwellian molecule, hard potential and hard sphere models. Combined with the results of Alonso et al. (Comm Math Phys 298: 293–322, 2010) for the soft potential and Maxwellian molecule models, we provide a unified form of \(L^p\) estimate for all cutoff models which are sharp in the sense of scaling. The most striking feature of our new estimates for the hard potential and hard sphere models is that they do not increase the moment, the same as Maxwellian molecule and soft potential models. Based on these novelties, we prove the global existence and scattering of the non-negative unique mild solution for the Cauchy problem of the Boltzmann equation when the positive initial data is small in the weighted \(L^3_{x,v}\) space.
期刊介绍:
The Archive for Rational Mechanics and Analysis nourishes the discipline of mechanics as a deductive, mathematical science in the classical tradition and promotes analysis, particularly in the context of application. Its purpose is to give rapid and full publication to research of exceptional moment, depth and permanence.